Disjunctive linear separation conditions and mixed-integer formulations for aircraft conflict resolution by speed and altitude control.

With the forecasted worldwide increase in travel volumes, it is a priority to improve the capacity of air traffic networks to minimize the economic cost of congestion and improve social welfare. In this paper, we introduce new mixed-integer programming formulations for aircraft conflict resolution with speed and altitude control which are based on disjunctive linear separation conditions that fully characterize the set of conflict-free aircraft trajectories. Two different objective function are proposed with piecewise linear and quadratic penalties, respectively, resulting in MILP and MIQP formulations. The performance of the proposed Disjunctive model is evaluated using benchmarking conflict resolution instances with up to 100 aircraft and 10 flight levels. Further, the proposed Disjunctive formulations are compared against alternative formulations based on existing and widely used separation constraints. Our results show that the proposed Disjunctive model outperforms existing formulations in the literature and can solve to optimality significantly more instances. Further, instances with up to 50 aircraft can be solved in a less than a second which highlights the potential of this approach as a decision-support tool for tactical conflict resolution.